f[x_,{y_,z_}]:={z,3*x*y}

y0=2

z0=-1

x0=1

g1[x_]:={y0,z0}+Integrate[f[t,{y0,z0}],{t,x0,x}]

MatrixForm[FullSimplify[g1[x]]]

g2[x_]:={y0,z0}+Integrate[f[t,g1[t]],{t,x0,x}]

MatrixForm[FullSimplify[g2[x]]]

g3[x_]:={y0,z0}+Integrate[f[t,g2[t]],{t,x0,x}]

MatrixForm[FullSimplify[g3[x]]]